Given the linear Anosov map $F_L$ of $\mathbb{T}^2$: $$z\mapsto \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix} z \quad\text{mod }1$$ it's already proved that there exists $P\in\mathbb N$ such that $F_L^P|_{M_N}=Id$, where $M_N$ denotes the set of points of the torus whose coordinates are rational with denominator $N\in\mathbb N$.
I want to prove that the minimum of such number is $P(N)$ and we have that $P(N)\leq N^4$. My ideas stop at the point where I note that any rational point with common denominator $N$ is a sum of the vectors $(1/N, 0)$ and $(0,1/N)$ with integer coefficients. Your help is much appreciated.